- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Extended framework for the hybrid Monte Carlo in lattice gauge theory

Norman H. Christ1, Lu-Chang Jin2, Christoph Lehner3, Erik Lundstrum1, and Nobuyuki Matsumoto4,*

  • *Contact author: nmatsum@bu.edu
PDF

Phys. Rev. D 112, 034507 – Published 25 August, 2025

DOI: https://doi.org/10.1103/k6dm-lj56

Abstract

We develop an extended framework for the hybrid Monte Carlo (HMC) algorithm in lattice gauge theory by embedding the SU(N) group into the space of general complex matrices, MN(C). Auxiliary directions will be completely factorized in the path integral, and the embedding does not alter the expectation values of the original theory. We perform the molecular dynamics updates by using the matrix elements of W∈MN(C) as the dynamical variables without group theoretic constraints. The framework enables us to introduce nonseparable Hamiltonians for the HMC in lattice gauge theory exactly, whose immediate application includes the Riemannian manifold HMC.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (32)

  1. S. Duane, A. D. Kennedy, B. J. Pendleton, and D. Roweth, Hybrid Monte Carlo, Phys. Lett. B 195, 216 (1987).
  2. G. Parisi, Prolegomena to any future computer evaluation of the QCD mass spectrum, in Progress in Gauge Field Theory, edited by G. ’t Hooft, A. Jaffe, H. Lehmann, P. K. Mitter, I. M. Singer, and R. Stora (Springer US, Boston, MA, 1984), pp. 531–541.
  3. G. G. Batrouni, G. R. Katz, A. S. Kronfeld, G. P. Lepage, B. Svetitsky, and K. G. Wilson, Langevin simulations of lattice field theories, Phys. Rev. D 32, 2736 (1985).
  4. C. T. H. Davies, G. G. Batrouni, G. R. Katz, A. S. Kronfeld, G. P. Lepage, P. Rossi, B. Svetitsky, and K. G. Wilson, Fourier acceleration in lattice gauge theories. 3. Updating field configurations, Phys. Rev. D 41, 1953 (1990).
  5. S. Duane, R. Kenway, B. J. Pendleton, and D. Roweth, Acceleration of gauge field dynamics, Phys. Lett. B 176, 143 (1986).
  6. S. Duane and B. J. Pendleton, Gauge invariant fourier acceleration, Phys. Lett. B 206, 101 (1988).
  7. M. Girolami and B. Calderhead, Riemann manifold Langevin and Hamiltonian Monte Carlo methods, J. R. Stat. Soc. Ser. B 73, 123 (2011).
  8. T. Nguyen, P. Boyle, N. H. Christ, Y.-C. Jang, and C. Jung, Riemannian manifold hybrid Monte Carlo in lattice QCD, Proc. Sci., LATTICE2021 (2022) 582.
  9. C. Jung and N. H. Christ, Riemannian manifold HMC with fermions, Proc. Sci., LATTICE2023 (2024) 009 [arXiv:2401.13226].
  10. G. Cossu, P. Boyle, N. Christ, C. Jung, A. Jüttner, and F. Sanfilippo, Testing algorithms for critical slowing down, EPJ Web Conf. 175, 02008 (2018).
  11. S. Fields, P. Boyle, N. Christ, C. Jung, and C. Lehner, Tuning the Riemannian manifold hybrid Monte Carlo with fermions, Proc. Sci., LATTICE2024 (2025) 045.
  12. C. T. H. Davies, G. G. Batrouni, G. R. Katz, A. S. Kronfeld, G. P. Lepage, K. G. Wilson, P. Rossi, and B. Svetitsky, Fourier acceleration in lattice gauge theories. 1. Landau gauge fixing, Phys. Rev. D 37, 1581 (1988).
  13. A. Sheta, Y. Zhao, and N. H. Christ, Gauge-fixed Fourier acceleration, Proc. Sci., LATTICE2021 (2022) 084 [arXiv:2108.05486].
  14. T. Cohen, M. Weiler, B. Kicanaoglu, and M. Welling, Gauge equivariant convolutional networks and the icosahedral CNN, in Proceedings of the 36th International Conference on Machine Learning, Proceedings of Machine Learning Research Vol. 97, edited by K. Chaudhuri and R. Salakhutdinov (PMLR, 2019), pp. 1321–1330.
  15. G. Kanwar, M. S. Albergo, D. Boyda, K. Cranmer, D. C. Hackett, S. Racanière, D. J. Rezende, and P. E. Shanahan, Equivariant flow-based sampling for lattice gauge theory, Phys. Rev. Lett. 125, 121601 (2020).
  16. M. Favoni, A. Ipp, D. I. Müller, and D. Schuh, Lattice gauge equivariant convolutional neural networks, Phys. Rev. Lett. 128, 032003 (2022).
  17. C. Lehner and T. Wettig, Gauge-equivariant neural networks as preconditioners in lattice QCD, Phys. Rev. D 108, 034503 (2023).
  18. C. Lehner and T. Wettig, Gauge-equivariant pooling layers for preconditioners in lattice QCD, Phys. Rev. D 110, 034517 (2024).
  19. Y. Nagai and A. Tomiya, Self-learning Monte Carlo with equivariant transformer, J. Phys. Soc. Jpn. 93, 114007 (2024).
  20. J. Sexton and D. Weingarten, Hamiltonian evolution for the hybrid Monte Carlo algorithm, Nucl. Phys. B380, 665 (1992).
  21. M. Luscher, and Trivializing maps, The Wilson flow, and the HMC algorithm, Commun. Math. Phys. 293, 899 (2010).
  22. N. Matsumoto, nobuyukimatsumoto68/gn (v1.1) (2025), 10.5281/zenodo.16809426.
  23. C. Lehner, M. Bruno, D. Richtmann, M. Schlemmer, R. Lehner, D. Knüttel, T. Wurm, L. Jin, S. Bürger, A. Hackl, and A. Klein, grid python toolkit (GPT), 2024-10 (2024), 10.5281/zenodo.14017415.
  24. I. P. Omelyan, I. M. Mryglod, and R. Folk, Construction of high-order force-gradient algorithms for integration of motion in classical and quantum systems, Phys. Rev. E 66, 026701 (2002).
  25. T. Takaishi, Choice of integrator in the hybrid Monte Carlo algorithm, Comput. Phys. Commun. 133, 6 (2000).
  26. T. Takaishi and P. de Forcrand, Testing and tuning symplectic integrators for the hybrid Monte Carlo algorithm in lattice QCD, Phys. Rev. E 73, 036706 (2006).
  27. Y.-F. Tang, Formal energy of a symplectic scheme for Hamiltonian systems and its applications (I), Comput. Math. Appl. 27, 31 (1994).
  28. A. D. Kennedy, P. J. Silva, and M. A. Clark, Shadow Hamiltonians, poisson brackets, and gauge theories, Phys. Rev. D 87, 034511 (2013).
  29. A. D. Kennedy and P. Rossi, Classical mechanics on group manifolds and applications to hybrid Monte Carlo, Nucl. Phys. B327, 782 (1989).
  30. K. S. Turitsyn, M. Chertkov, and M. Vucelja, Irreversible Monte Carlo algorithms for efficient sampling, Physica (Amsterdam) 240D, 410 (2011).
  31. P. A. Boyle, G. Cossu, A. Yamaguchi, and A. Portelli, Grid: A next generation data parallel c++ QCD library, Proc. Sci., LATTICE2015 (2016) 023.
  32. A. Yamaguchi, P. Boyle, G. Cossu, G. Filaci, C. Lehner, and A. Portelli, Grid: OneCode and FourAPIs, Proc. Sci., LATTICE2021 (2022) 035.

Outline

Information

Phys. Rev. D 112, 034507– Published 25 August, 2025
  • Received 16 January 2025
  • Accepted 30 July 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation